In this section we state some results which were proved in [1]}, where the authors study some properties of homeomorphisms of the 2-sphere with no topological horseshoes. A compact subset \(Y\)  of a manifold \(M\)  is a topological horseshoe if it is invariant by an iterated \(f^q\)  of \(f\) , \(f^q\vert _Y\)  is semi-conjugated to \(g:Z\rightarrow Z\)  on a Hausdorff compact space, so that the fibers by the factor map are all finite with an uniform bound \(m\)  in their cardinality, and \(g\)  is semi-conjugated to the Bernoulli shift \(\sigma :\lbrace 1,\cdots ,N\rbrace \rightarrow \lbrace 1,\cdots ,N\rbrace \) , where \(N\ge 2\)  is such that the preimage of every \(s\) -periodic sequence of \(\lbrace 1,\cdots ,N\rbrace \)  by the factor map contains a \(s\) -periodic point of \(g\) .
